**Note1:**Today, math is taught for test-based proficiency, not mastery, which hurts all kids, especially children of color. Consequently, students lack good math skills in basic arithmetic and algebra. Weak math skills limit career choices later on because math is the cornerstone of technology, science, and engineering.

**Kids need to**

**drill-to-develop-skill****and master the content fundamentals**, but this is not the focus in reform math classrooms.

*Parents should give their children extra math practice daily.*Learning the multiplication table isn't always fun. It is hard work! Students should memorize single-digit number facts to focus on compact,

**standard algorithms**.

Unfortunately, educators are teaching reform math instead of traditional arithmetic. The reform curriculum is wrong for beginners.

**Kids are novices, not pint-sized mathematicians.**

**Note2:**Kids who lack specific knowledge can't do critical thinking well. Critical thinking in math is different from critical thinking in science or from critical thinking in literature, and so on.

**Critical thinking is not a generalized skill; it is domain specific!**

*Kids cannot solve math problems unless they know the math.*Knowing is remembering from

**long-term memory.**

*Lastly,*a

*lways remember that i*

*nnovation comes from knowledge, not thin air.***Read my**

__New Science Page__.The Next Gen science standards are not science standards, they are STEM standards, but they fall flat.

*They shortchange science by limiting core content*; consequently, they do not prepare high school students for college-level chemistry and physics courses or STEM fields such as engineering and computer science. (Note: AP calculus is not accepted toward a STEM major at some universities.)

**Aside:**If you major in the humanities, then be sure to take a lot of math and science, too, such as calculus and physics.

**Pre-Algebra Course**

Before the Common Core era, many 7th-grade students studied pre-algebra with trig. Most major Algebra-1 topics were examined in the pre-algebra texts such as Glencoe's

*Pre-Algebra*, but that was the 1990s and 2000s. The best pre-algebra textbooks also covered right-triangle trig ratios for indirect measurements. Instead of using trig tables, which were not in the book, students used a scientific calculator to solve for

*x*in trig equations such as

**tan 5.2º =**.

*x*/55**A thorough pre-algebra course trains students for Algebra-1 in 8th grade.**

The primary recommendation of the National Math Advisory Panel (2008) was to get more students ready for Algebra-1 by 8th grade.

*Common Core ignored it*.

Students coming into 7th grade should attain a particular skill set to be ready for a pre-algebra course, but Common Core doesn't get them to that level. In fact, there is no "pre-algebra" course in Common Core. Also, starting in the 1st grade, the Common Core and state math standards are not world-class mathematics. Kids are not learning much arithmetic because some topics are delayed, taught poorly, or not taught at all. Furthermore, under the yoke of reform math, the fundamentals are not practiced for mastery. Math education has not been effective or efficient for many students. Efforts to make instruction and learning more efficient didn't improve achievement, that is, smaller class size, personalized learning, and other fads fell flat.

**My Contrarian Math Page**

Online credit recovery, grade inflation, and watered-down courses have contributed to bogus high-school graduation rates. What a scam! 75% of high school seniors are not proficient in mathematics (NAEP). Also, strict schooling does not dent a child's curiosity, creativity, or motivation. Think, Einstein. "In education, you increase differences." Think, Feynman.

In math, knowing facts and compact procedures (e.g., standard algorithms) in long-term memory builds the foundation for understanding, applying, and reasoning in math.

Learning is remembering from long-term memory!

Zig Engelmann points out, "Without a firm foundation in number facts [automated in long-term memory], children are held back from further learning."

Model (Knowing facts and procedures....): GabrielaB

**Students cannot use something they don't know well in long-term memory.**Indeed, the memorization of essential facts and efficient, compact procedures (i.e., the standard algorithms) play a vital role in performing math at an acceptable level.***Standard arithmetic is simple and compact, but it is not easy to learn without the memorization of single-digit number facts and practice-practice-practice. ***The primary learning goal should be the mastery of necessary content in long-term memory, not proficiency on state tests. But, it's not!******Learning is remembering from long-term memory!

*"You don't know anything until you have practiced." (Richard Feynman)*

Zig Engelmann points out, "Without a firm foundation in number facts [automated in long-term memory], children are held back from further learning."

Model (Knowing facts and procedures....): GabrielaB

Memorizing The Times TableLearning the multiplication table in the 2nd and 3rd grades may not be much fun, but it is absolutely necessary to move forward in arithmetic. Automating the single-digit number facts in long-term memory is required for fluency in using the standard algorithms. Memorizing is not without some level of understanding (i.e., meaning). For example, students should be taught that 3 x 4 means the sum of three fours: 4 + 4 + 4 = 12 (i.e., repeated addition). Also 4 x 3 means the sum of four threes: 3 + 3 + 3 + 3 = 12 (repeated addition). Both give the same answer because multiplication, like addition, is commutative: ab = ba.Note: Singapore 1st-grade students start multiplication as repeated addition, memorize half the multiplication facts in 2nd grade and the rest in 3rd grade. |

**How Before Why**

Tobias Dantzig (

*Number: The Language of Science*), a book highly praised by Albert Einstein, writes, "In the history of mathematics, the "how" always preceded the "why," the technique of the subject preceded its philosophy. This is particularly true of arithmetic. The strength of arithmetic lies in its absolute generality. a + b = b + a. Its rules admit of no exceptions: they apply to all numbers. Every number has a successor [add one]. There is an infinity of numbers."

American teachers are hung up on the "why" of everything, but in arithmetic, it is best to learn the "how" first and the "why" later.

*Learn the technique first and get it right (i,e., automate it).*Note: Standard arithmetic is simple and compact, but it is not easy to learn without memorization of single-digit number facts and practice-practice-practice.

**Measure & Calculate**

Mr. Wizard (Don Herbert) gave children instruction and direction. "The next time you want to be accurate about something, make no judgments. Instead, measure and calculate."

I prepared for college without phonics, reform math, a graphing calculator, screens, a cell phone, the Internet, or group work. When I was interested in something, I read books. I studied. I learned on my own.

**Quality Math Instruction**

Gina Picha is a typical reform-math ideologue. In

*Education Week*she wrote, "So what is quality math instruction? Quality math instruction is real-world, collaborative, and involves productive struggle, debate, and conversation. Integrating unproductive math practices into other content areas is counterproductive for any STEM program or school." Really?

**Her vision of quality math instruction is radical, fuzzy, and doesn't work. It is not the way to teach novices the fundamentals of arithmetic.**

*Mathematics is part of STEM, even in elementary school.*

**STEM Requires Math Skills, Even in Elementary School**

Traditional arithmetic instruction, which Picha wrongly describes as unproductive math practices, does not screw up STEM. In fact,

*Science A Process Approach*1967 for K-6 successfully integrated traditional math skills with science skills in elementary school using a hierarchical approach of prerequisites (Gagne) through clearly defined behavioral objectives (Mager). Regrettably, SAPA failed because teachers didn't know enough math or science to teach the curriculum well. SAPA required strong teacher guidance during instruction. Professional Development for SAPA didn't work in the late 60s and early 70s. PD doesn't work today, either, not for teachers who majored in education. Education should not be a major.

*Teachers should major in an academic subject.*

**Barrier to STEM**

The barrier to STEM and higher-level math has always been weak math skills, that is, students lacked sufficient knowledge of standard arithmetic and algebra (skills, ideas, and uses). Often, reform math programs do not stress the mastery of essential content in long-term memory, which requires practice-practice-practice. Moreover, it is traditional or standard arithmetic, not reform math, that primes a child's mind for higher math and problem-solving (i.e., critical thinking in math).

Note: Gee, I always thought that math was not opinion.

**NAEP: We teach math poorly!**

The proficiency standards of the National Assessment of Educational Progress (NAEP), the Nation's Report Card, show what students “should know and be able to do.” Many say the proficiency benchmarks, which were adopted in 1992, are misleading and nothing more than the opinion or judgment of a panel. Some say the criteria are set too high. I diverge.

*The committee has changed hands several times, but the proficiency benchmarks, which are provisional, were never adjusted.*

**To me, if 60% of the 4th graders, 66% of 8th graders, and 75% of 12th graders can't do math well, then we have a huge problem.**Misleading? Hardly! It seems clear that U.S. educators have been teaching math poorly, which is nothing new. It is the status quo of progressivism [aka liberal ideology], which has been hard to change.

*The problem is the way we teach math--both content and methods.*Simply put, the obstacles have been the "teaching" (Zig Engelmann) and the "fallacies of fairness" (Thomas Sowell), which mix high-achieving students with low-achieving students in K-8 classrooms and equalize downward.

**Zig Engelmann**points out, "Without a firm foundation in number facts [automated in long-term memory], children are held back from further learning."

*I figured out that reform math via Common Core or state standards has been the wrong path to teach young children basic arithmetic.*

**Community College Remedial Math**

A red flag that math is taught poorly has been the high percentages of high-school graduates who end up in remedial math courses at community colleges. For example, of the nine school districts that feed students into the local community college (Tucson area), the range of students placed in remedial math is from 74% to 88% depending on the school district.

*The stumbling block has been*algebra

*, but difficulty with algebra implies that students have not mastered basic arithmetic, especially fractions. In short, basic or standard arithmetic is not taught well starting in 1st grade.*

The solution has not been to fix the inadequate teaching of arithmetic so that students are prepared for algebra no later than the 8th grade but to downplay algebra as a requirement for community college entry. Richard Webb, Editor,

*New Scientist: How Numbers Work*, writes, "Algebra allows us to represent and manipulate knowledge in terms of numbers, symbols, and equations, and as such is the broadest pillar of formal higher mathematics." Standard arithmetic is simple, but it is not easy to learn without memorization and practice-practice-practice.

**Most students don't know what they should know!**

Still, we have people announcing to parents, teachers, and citizens not to be alarmed by the high percentages of students who do not meet the NAEP proficiency benchmarks in the 4th and 8th grades. Really? The benchmarks are not bogus or set too high. The NAEP scores show that math is poorly taught. The inadequate teaching of standard arithmetic doesn't begin in middle school; it starts in 1st grade.

*Also, American students are bad at fractions, even with a calculator.*[Note: NAEP government tests (aka The Nation's Report Card) are given every two years to 4th and 8th-grade students in every state. The 2017 results were released on April 10, 2018. Incidentally, the members of the NAEP panel change every few years. While the panels change, the benchmarks, which are

*provisional*, have not been changed since adopted. It seems clear that after all these years most students still don't know what they should know, which is a "teaching" problem. We should stop giving excuses for lackluster performance.] Also: read

__Math Teaching.__

**Fundamentals-First Recipe**

We don't need radical changes and reforms driven by fads in math teaching; however, starting in 1st grade, students do need to learn standard [traditional] arithmetic--the fundamentals--for mastery, not [NCTM] reform math for test proficiency. The "fundamentals-first" recipe for success goes against the education establishment, but it is not radical; it is common sense based on the cognitive science of learning. In short, we should focus on a "fundamentals-first" means of achievement and a well-thought-out, consistent curriculum in mathematics. The current curriculum is not coherent; it is fragmented and warped by one-size-fits-all state tests.

*The liberal ideology is that all students are equal and, therefore, should receive the same instruction. The assumption is flat wrong. The fact is that students vary widely in academic ability and achievement. They are not all equal.*

**Critical Thinking**

Daniel T. Willingham, a cognitive scientist, points out, "Knowledge is critical to thought."

He observes, "Critical thinking in one domain does not apply to another." For example, problem-solving or critical thinking in mathematics is not relevant to chemistry or history, etc. Immanuel Kant (

*Critique of Pure Reason)*remarked, "Thoughts without content are empty."

*Be skeptical of those who claim they teach critical thinking independent of knowledge.*

**The Best Preparation for the Future**

Knowledge has always been the best preparation for the future no matter the epoch, including the booming Atomic (Quantum), Space, Communication, Computer-Internet, and AI eras. Furthermore, the importance of math has endured and soared over the years. Kids need more mathematical knowledge than before, both factual and procedural, to prepare for the future, not less.

*We should teach basic math for mastery, not merely for test-based proficiency. Moreover, the state test should measure what is taught in the classroom, not vice versa.*

**Click & Read:**

__Future.__<--Caption: A student works in the

If learning is remembering from long-term memory, then as Zig Engelmann points out, "You learn only through mastery" (i.e., practice-practice-practice). Unlike American elementary schools, students in other nations such as Russia learn the standard algorithms for multiplication (e.g., 4987 x 6) and long division (e.g., 4987 ÷ 8) no later than the 3rd grade through practice-practice-practice.

In the real world, 54% of Singapore 8th-grade students scored at the Advanced Level compare to only 10% of American 8th-grade students (TIMSS). Note: The majority of students who want to go to college will likely end up in remedial math because they have not mastered basic arithmetic and algebra. Children with weak math skills have limited career opportunities later on. A student does not need to be gifted, or a genius to learn Algebra-1 in middle school, AP Calculus in high school, or grasp some algebra fundamentals in 1st grade, but they have to be prepared. Average kids can do these things when they are taught well and work hard to achieve.

< Model (Kumon Pre-Algebra): GabbyB

*Kumon Pre-Algebra* practice book.*Unlike reform math, Kumon math focuses on the mastery of mechanics first, and it works.***Zig Engelmann**If learning is remembering from long-term memory, then as Zig Engelmann points out, "You learn only through mastery" (i.e., practice-practice-practice). Unlike American elementary schools, students in other nations such as Russia learn the standard algorithms for multiplication (e.g., 4987 x 6) and long division (e.g., 4987 ÷ 8) no later than the 3rd grade through practice-practice-practice.

**Advanced Level of TIMSS**In the real world, 54% of Singapore 8th-grade students scored at the Advanced Level compare to only 10% of American 8th-grade students (TIMSS). Note: The majority of students who want to go to college will likely end up in remedial math because they have not mastered basic arithmetic and algebra. Children with weak math skills have limited career opportunities later on. A student does not need to be gifted, or a genius to learn Algebra-1 in middle school, AP Calculus in high school, or grasp some algebra fundamentals in 1st grade, but they have to be prepared. Average kids can do these things when they are taught well and work hard to achieve.

< Model (Kumon Pre-Algebra): GabbyB

**Comment: Asian System**

The Asian system is built on memorization, which forces students to store information in long-term memory where it is ready for use to solve problems. American educators don't get it. Asian children are taught mechanics first with the explanation later, and it works!

*We do it backward with understanding first, and it doesn't work. In short, the progressive math reforms have not stressed the mastery of standard arithmetic in long-term memory.*

**"In education**

**, you increase differences."**

Richard P. Feynman was invited to a conference to discuss "the ethics of equality in education." He confronted the experts by asking this question. "In education, you increase differences. If someone's good at something, you try to develop his ability, which results in differences, or inequalities. So if education increases inequality, is this ethical?"

**(**

*Surely You're Joking, Mr. Feynman!*by Richard P. Feynman, Nobel Prize in Physics

**)**.

"You don't know anything until you have practiced." (Feynman)

**Practice does not create talent.**There has to be something there, to begin with, explains Ian Stewart, mathematician. Talent doesn't come from thin air. Practice does not make perfect; it improves a skill according to your ability, the quality of instruction, practice habits, and persistence.

**Children Vary Widely**

School children vary widely in academic ability, musical ability, athletic ability, writing ability, language ability, etc.

*That said, most children can learn arithmetic and algebra at an acceptable level, but they have to work at it; i.e., drill to improve skill. The problem I see is that state testing seems to drive what is taught in the classroom, which is to answer typical state test questions.*Parents still don't know if their 3rd graders can do long division or their 4th graders can divide fractions, which are essential skills to get ready for Algebra. Kids should know the standard algorithms for addition, subtraction, multiplication, and division by the end of 3rd grade, but they don't. If learning is remembering from long-term memory, then how can students remember something that wasn't taught and practiced sufficiently to stick in long-term memory? "You don't know anything until you have practiced."

****

**©2004-2018**LT/ThinkAlgebra

Model Credits: ShaynaT, SierahG, Alyssa, McKaylaS, KaileyP, ChloeM, GrahamB, SierahG, HannahE, KrystalV, RemiB, GabbyB, and others. Most of the models were referred to me by Barbizon.

Contact: LarryT -- ThinkAlgebra@cox.net

Changes made: 7-18-17, 8-12-17, 8-20-17, 8-28-17, 9-8-17, 9-14-17, 9-23-17, 10-1-17, 10-2-17, 10-10-17, 10-20-17, 11-10-17, 11-18-17, 12-5-17, 12-23-17, 3-26-18, 5-5-18, 6-8-18, 6-22-18, 7-12-18, 7-18-18, 7-24-18, 8-11-18, 9-8-18